180 ON THE COLOURS OF THICK PLATES.
The formula (27), (28), and (29) shew that X is equal to the semi-sum of £ and £, and for the same reason Y is equal to the semi-sum of?? and ?;x. Hence the geometrical construction given in Art. 11 for finding the centre of the system in the case of a plane mirror applies equally to a curved mirror, even when the curvatures of the two surfaces are different. Since the retardation vanishes for the image itself, it follows that the achromatic line is a circle having the two points of intersection above mentioned for opposite extremities of a diameter.
20. It follows from the expressions for X and F, or from the geometrical construction to which they lead, that if the eye be not in the line joining the luminous point and its image, whenever it crosses either of two planes drawn perpendicular to the axis, and passing, one through the luminous point, and the other through its image, the centre of curvature of the bands moves off to an infinite distance, and the bands become straight, and then bend round the other way.
When the eye coincides with the luminous point, /, g, h become equal to a, b, c, respectively, and R vanishes independently of x and y. The same takes place when the eye coincides with the image, since in this case
f a g & 1 1 2
./__.. __ — = __ __ I __ r— —
h c ' h c ' h c p '
Hence, when the eye crosses either of the planes above mentioned, remaining in the line joining the luminous point and its image, instead of bands which become straight and then change curvature, we have rings which disappear by moving off to infinity, and then appear again.
I have verified these conclusions by experiment, substituting when necessary a virtual image of the eye for the eye itself, in the manner explained in Art. 15. The experiments embraced the following cases, in the description of which 0 will be used to